The blasts were heard across the city and plumes of smoke were seen rising from the depot, where looting by residents had broken out a day earlier, an AFP correspondent reported.
Houses shook, many windows were shattered and several nearby buildings were destroyed after the depot blasts, residents said.
The depot is located at the foot of Jabal Hadid, a mountain near the port of Aden.
A large amount of weapons dating back to the Soviet era were stored in a cave.
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Many people were in the depot or climbing on Jabal Hadid when the explosions took place, said witnesses.
No casualty toll was immediately available.
President Abedrabbo Mansour Hadi took refuge in Aden after fleeing the rebel-held capital last month.
He left the southern port city this week and arrived in Saudi Arabia before heading to Egypt for a regional summit expected to be dominated by Arab air strikes on the Shiite Huthi rebels in Yemen.